3.21.54 \(\int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} (e^2 \log (5)+e^{2+x} (3+x) \log (5))}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx\)

Optimal. Leaf size=30 \[ \frac {e^2 \log (5)}{5+e^{\frac {e^{e^x}}{-1+\frac {x}{3+x}}}+x} \]

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Rubi [F]  time = 44.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*E^2*Log[5] + E^(E^x + (E^E^x*(-3 - x))/3)*(E^2*Log[5] + E^(2 + x)*(3 + x)*Log[5]))/(75 + 3*E^((2*E^E^x
*(-3 - x))/3) + 30*x + 3*x^2 + E^((E^E^x*(-3 - x))/3)*(30 + 6*x)),x]

[Out]

(E^2*Log[5])/(5 + x) - (E^2*Log[5]*Defer[Int][E^(E^x + x)/(1 + E^((E^E^x*(3 + x))/3)*(5 + x))^2, x])/3 - E^2*L
og[5]*Defer[Int][1/((5 + x)^2*(1 + E^((E^E^x*(3 + x))/3)*(5 + x))^2), x] - (E^2*Log[5]*Defer[Int][E^E^x/((5 +
x)*(1 + E^((E^E^x*(3 + x))/3)*(5 + x))^2), x])/3 + (2*E^2*Log[5]*Defer[Int][E^(E^x + x)/((5 + x)*(1 + E^((E^E^
x*(3 + x))/3)*(5 + x))^2), x])/3 + (E^2*Log[5]*Defer[Int][E^(E^x + x)/(1 + E^((E^E^x*(3 + x))/3)*(5 + x)), x])
/3 + 2*E^2*Log[5]*Defer[Int][1/((5 + x)^2*(1 + E^((E^E^x*(3 + x))/3)*(5 + x))), x] + (E^2*Log[5]*Defer[Int][E^
E^x/((5 + x)*(1 + E^((E^E^x*(3 + x))/3)*(5 + x))), x])/3 - (2*E^2*Log[5]*Defer[Int][E^(E^x + x)/((5 + x)*(1 +
E^((E^E^x*(3 + x))/3)*(5 + x))), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^2 \left (-3+e^{e^x-\frac {1}{3} e^{e^x} (3+x)}+e^{e^x+x-\frac {1}{3} e^{e^x} (3+x)} (3+x)\right ) \log (5)}{3 \left (5+e^{-\frac {1}{3} e^{e^x} (3+x)}+x\right )^2} \, dx\\ &=\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {-3+e^{e^x-\frac {1}{3} e^{e^x} (3+x)}+e^{e^x+x-\frac {1}{3} e^{e^x} (3+x)} (3+x)}{\left (5+e^{-\frac {1}{3} e^{e^x} (3+x)}+x\right )^2} \, dx\\ &=\frac {1}{3} \left (e^2 \log (5)\right ) \int \left (-\frac {3}{(5+x)^2}-\frac {3+5 e^{e^x}+15 e^{e^x+x}+e^{e^x} x+8 e^{e^x+x} x+e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {6+5 e^{e^x}+15 e^{e^x+x}+e^{e^x} x+8 e^{e^x+x} x+e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}\right ) \, dx\\ &=\frac {e^2 \log (5)}{5+x}-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {3+5 e^{e^x}+15 e^{e^x+x}+e^{e^x} x+8 e^{e^x+x} x+e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {6+5 e^{e^x}+15 e^{e^x+x}+e^{e^x} x+8 e^{e^x+x} x+e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx\\ &=\frac {e^2 \log (5)}{5+x}-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {3+e^{e^x} (5+x)+e^{e^x+x} \left (15+8 x+x^2\right )}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {6+e^{e^x} (5+x)+e^{e^x+x} \left (15+8 x+x^2\right )}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx\\ &=\frac {e^2 \log (5)}{5+x}-\frac {1}{3} \left (e^2 \log (5)\right ) \int \left (\frac {3}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {5 e^{e^x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {15 e^{e^x+x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {e^{e^x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {8 e^{e^x+x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}+\frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2}\right ) \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \left (\frac {6}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}+\frac {5 e^{e^x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}+\frac {15 e^{e^x+x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}+\frac {e^{e^x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}+\frac {8 e^{e^x+x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}+\frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )}\right ) \, dx\\ &=\frac {e^2 \log (5)}{5+x}-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx-\left (e^2 \log (5)\right ) \int \frac {1}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx-\frac {1}{3} \left (5 e^2 \log (5)\right ) \int \frac {e^{e^x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx+\frac {1}{3} \left (5 e^2 \log (5)\right ) \int \frac {e^{e^x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx+\left (2 e^2 \log (5)\right ) \int \frac {1}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx-\frac {1}{3} \left (8 e^2 \log (5)\right ) \int \frac {e^{e^x+x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx+\frac {1}{3} \left (8 e^2 \log (5)\right ) \int \frac {e^{e^x+x} x}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx-\left (5 e^2 \log (5)\right ) \int \frac {e^{e^x+x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )^2} \, dx+\left (5 e^2 \log (5)\right ) \int \frac {e^{e^x+x}}{(5+x)^2 \left (1+5 e^{\frac {1}{3} e^{e^x} (3+x)}+e^{\frac {1}{3} e^{e^x} (3+x)} x\right )} \, dx\\ &=\frac {e^2 \log (5)}{5+x}-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x} x}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx-\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x} x}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx+\frac {1}{3} \left (e^2 \log (5)\right ) \int \frac {e^{e^x+x} x^2}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx-\left (e^2 \log (5)\right ) \int \frac {1}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx-\frac {1}{3} \left (5 e^2 \log (5)\right ) \int \frac {e^{e^x}}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx+\frac {1}{3} \left (5 e^2 \log (5)\right ) \int \frac {e^{e^x}}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx+\left (2 e^2 \log (5)\right ) \int \frac {1}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx-\frac {1}{3} \left (8 e^2 \log (5)\right ) \int \frac {e^{e^x+x} x}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx+\frac {1}{3} \left (8 e^2 \log (5)\right ) \int \frac {e^{e^x+x} x}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx-\left (5 e^2 \log (5)\right ) \int \frac {e^{e^x+x}}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )^2} \, dx+\left (5 e^2 \log (5)\right ) \int \frac {e^{e^x+x}}{(5+x)^2 \left (1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.21, size = 41, normalized size = 1.37 \begin {gather*} \frac {e^{2+\frac {1}{3} e^{e^x} (3+x)} \log (5)}{1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*E^2*Log[5] + E^(E^x + (E^E^x*(-3 - x))/3)*(E^2*Log[5] + E^(2 + x)*(3 + x)*Log[5]))/(75 + 3*E^((2
*E^E^x*(-3 - x))/3) + 30*x + 3*x^2 + E^((E^E^x*(-3 - x))/3)*(30 + 6*x)),x]

[Out]

(E^(2 + (E^E^x*(3 + x))/3)*Log[5])/(1 + E^((E^E^x*(3 + x))/3)*(5 + x))

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fricas [A]  time = 0.56, size = 39, normalized size = 1.30 \begin {gather*} \frac {e^{\left (e^{x} + 2\right )} \log \relax (5)}{{\left (x + 5\right )} e^{\left (e^{x}\right )} + e^{\left (-\frac {1}{3} \, {\left ({\left (x + 3\right )} e^{\left (e^{x} + 2\right )} - 3 \, e^{\left (x + 2\right )}\right )} e^{\left (-2\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3*(-3-x)*exp(exp(x)))-3*exp(2)*log(5))
/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x+30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm="fricas")

[Out]

e^(e^x + 2)*log(5)/((x + 5)*e^(e^x) + e^(-1/3*((x + 3)*e^(e^x + 2) - 3*e^(x + 2))*e^(-2)))

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giac [B]  time = 1.48, size = 2069, normalized size = 68.97 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3*(-3-x)*exp(exp(x)))-3*exp(2)*log(5))
/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x+30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm="giac")

[Out]

(x^5*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 21*x^4*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) +
2)*log(5) + 2*x^4*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + x^4*e^(3*x + 2*e^x + 2)*log(5) + 174*
x^3*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 36*x^3*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2
)*log(5) + 6*x^3*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) + x^3*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^
x) + 2)*log(5) + 16*x^3*e^(3*x + 2*e^x + 2)*log(5) + 2*x^3*e^(2*x + 2*e^x + 2)*log(5) + 710*x^2*e^(1/3*x*e^(e^
x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 240*x^2*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 78*x^2
*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) + 15*x^2*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5
) + 6*x^2*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x) + 2)*log(5) + 94*x^2*e^(3*x + 2*e^x + 2)*log(5) + 26*x^2*e^(2*x
+ 2*e^x + 2)*log(5) + 6*x^2*e^(2*x + e^x + 2)*log(5) + x^2*e^(x + 2*e^x + 2)*log(5) + 1425*x*e^(1/3*x*e^(e^x)
+ 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 700*x*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 330*x*e^(1
/3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) + 75*x*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5) + 60*
x*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x) + 2)*log(5) + 9*x*e^(1/3*x*e^(e^x) + x + e^(e^x) + 2)*log(5) + 240*x*e^
(3*x + 2*e^x + 2)*log(5) + 110*x*e^(2*x + 2*e^x + 2)*log(5) + 48*x*e^(2*x + e^x + 2)*log(5) + 10*x*e^(x + 2*e^
x + 2)*log(5) + 6*x*e^(x + e^x + 2)*log(5) + 1125*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 750*e
^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 450*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) +
125*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5) + 150*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x) + 2)*log(5)
+ 45*e^(1/3*x*e^(e^x) + x + e^(e^x) + 2)*log(5) + 225*e^(3*x + 2*e^x + 2)*log(5) + 150*e^(2*x + 2*e^x + 2)*lo
g(5) + 90*e^(2*x + e^x + 2)*log(5) + 25*e^(x + 2*e^x + 2)*log(5) + 30*e^(x + e^x + 2)*log(5) + 9*e^(x + 2)*log
(5))/(x^6*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 26*x^5*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 2*x^5
*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x)) + 2*x^5*e^(3*x + 2*e^x) + 279*x^4*e^(1/3*x*e^(e^x) + 3*x + 2*e^x +
e^(e^x)) + 46*x^4*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x)) + 6*x^4*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x)) +
x^4*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x)) + x^4*e^(-1/3*x*e^(e^x) + 3*x + 2*e^x - e^(e^x)) + 42*x^4*e^(3*x +
2*e^x) + 4*x^4*e^(2*x + 2*e^x) + 1580*x^3*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 420*x^3*e^(1/3*x*e^(e^x
) + 2*x + 2*e^x + e^(e^x)) + 108*x^3*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x)) + 20*x^3*e^(1/3*x*e^(e^x) + x + 2
*e^x + e^(e^x)) + 6*x^3*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x)) + 16*x^3*e^(-1/3*x*e^(e^x) + 3*x + 2*e^x - e^(e^
x)) + 2*x^3*e^(-1/3*x*e^(e^x) + 2*x + 2*e^x - e^(e^x)) + 348*x^3*e^(3*x + 2*e^x) + 72*x^3*e^(2*x + 2*e^x) + 12
*x^3*e^(2*x + e^x) + 2*x^3*e^(x + 2*e^x) + 4975*x^2*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 1900*x^2*e^(1/
3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x)) + 720*x^2*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x)) + 150*x^2*e^(1/3*x*e^(e
^x) + x + 2*e^x + e^(e^x)) + 90*x^2*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x)) + 9*x^2*e^(1/3*x*e^(e^x) + x + e^(e^
x)) + 94*x^2*e^(-1/3*x*e^(e^x) + 3*x + 2*e^x - e^(e^x)) + 26*x^2*e^(-1/3*x*e^(e^x) + 2*x + 2*e^x - e^(e^x)) +
6*x^2*e^(-1/3*x*e^(e^x) + 2*x + e^x - e^(e^x)) + x^2*e^(-1/3*x*e^(e^x) + x + 2*e^x - e^(e^x)) + 1420*x^2*e^(3*
x + 2*e^x) + 480*x^2*e^(2*x + 2*e^x) + 156*x^2*e^(2*x + e^x) + 30*x^2*e^(x + 2*e^x) + 12*x^2*e^(x + e^x) + 825
0*x*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 4250*x*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x)) + 2100*x*e^(1
/3*x*e^(e^x) + 2*x + e^x + e^(e^x)) + 500*x*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x)) + 450*x*e^(1/3*x*e^(e^x) +
x + e^x + e^(e^x)) + 90*x*e^(1/3*x*e^(e^x) + x + e^(e^x)) + 240*x*e^(-1/3*x*e^(e^x) + 3*x + 2*e^x - e^(e^x))
+ 110*x*e^(-1/3*x*e^(e^x) + 2*x + 2*e^x - e^(e^x)) + 48*x*e^(-1/3*x*e^(e^x) + 2*x + e^x - e^(e^x)) + 10*x*e^(-
1/3*x*e^(e^x) + x + 2*e^x - e^(e^x)) + 6*x*e^(-1/3*x*e^(e^x) + x + e^x - e^(e^x)) + 2850*x*e^(3*x + 2*e^x) + 1
400*x*e^(2*x + 2*e^x) + 660*x*e^(2*x + e^x) + 150*x*e^(x + 2*e^x) + 120*x*e^(x + e^x) + 18*x*e^x + 5625*e^(1/3
*x*e^(e^x) + 3*x + 2*e^x + e^(e^x)) + 3750*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x)) + 2250*e^(1/3*x*e^(e^x) +
2*x + e^x + e^(e^x)) + 625*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x)) + 750*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x)
) + 225*e^(1/3*x*e^(e^x) + x + e^(e^x)) + 225*e^(-1/3*x*e^(e^x) + 3*x + 2*e^x - e^(e^x)) + 150*e^(-1/3*x*e^(e^
x) + 2*x + 2*e^x - e^(e^x)) + 90*e^(-1/3*x*e^(e^x) + 2*x + e^x - e^(e^x)) + 25*e^(-1/3*x*e^(e^x) + x + 2*e^x -
e^(e^x)) + 30*e^(-1/3*x*e^(e^x) + x + e^x - e^(e^x)) + 9*e^(-1/3*x*e^(e^x) + x - e^(e^x)) + 2250*e^(3*x + 2*e
^x) + 1500*e^(2*x + 2*e^x) + 900*e^(2*x + e^x) + 250*e^(x + 2*e^x) + 300*e^(x + e^x) + 90*e^x)

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maple [A]  time = 0.08, size = 20, normalized size = 0.67

method	result	size
risch	\(\frac {{\mathrm e}^{2} \ln \relax (5)}{5+{\mathrm e}^{-\frac {\left (3+x \right ) {\mathrm e}^{{\mathrm e}^{x}}}{3}}+x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3+x)*exp(2)*ln(5)*exp(x)+exp(2)*ln(5))*exp(exp(x))*exp(1/3*(-3-x)*exp(exp(x)))-3*exp(2)*ln(5))/(3*exp(1
/3*(-3-x)*exp(exp(x)))^2+(6*x+30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x,method=_RETURNVERBOSE)

[Out]

exp(2)*ln(5)/(5+exp(-1/3*(3+x)*exp(exp(x)))+x)

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maxima [A]  time = 0.59, size = 34, normalized size = 1.13 \begin {gather*} \frac {e^{\left (\frac {1}{3} \, x e^{\left (e^{x}\right )} + e^{\left (e^{x}\right )} + 2\right )} \log \relax (5)}{{\left (x + 5\right )} e^{\left (\frac {1}{3} \, x e^{\left (e^{x}\right )} + e^{\left (e^{x}\right )}\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3*(-3-x)*exp(exp(x)))-3*exp(2)*log(5))
/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x+30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm="maxima")

[Out]

e^(1/3*x*e^(e^x) + e^(e^x) + 2)*log(5)/((x + 5)*e^(1/3*x*e^(e^x) + e^(e^x)) + 1)

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mupad [B]  time = 1.26, size = 23, normalized size = 0.77 \begin {gather*} \frac {{\mathrm {e}}^2\,\ln \relax (5)}{x+{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{3}}+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(2)*log(5) - exp(exp(x))*exp(-(exp(exp(x))*(x + 3))/3)*(exp(2)*log(5) + exp(2)*exp(x)*log(5)*(x + 3
)))/(30*x + 3*exp(-(2*exp(exp(x))*(x + 3))/3) + exp(-(exp(exp(x))*(x + 3))/3)*(6*x + 30) + 3*x^2 + 75),x)

[Out]

(exp(2)*log(5))/(x + exp(- exp(exp(x)) - (x*exp(exp(x)))/3) + 5)

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sympy [A]  time = 0.30, size = 22, normalized size = 0.73 \begin {gather*} \frac {e^{2} \log {\relax (5 )}}{x + e^{\left (- \frac {x}{3} - 1\right ) e^{e^{x}}} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+x)*exp(2)*ln(5)*exp(x)+exp(2)*ln(5))*exp(exp(x))*exp(1/3*(-3-x)*exp(exp(x)))-3*exp(2)*ln(5))/(3
*exp(1/3*(-3-x)*exp(exp(x)))**2+(6*x+30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x**2+30*x+75),x)

[Out]

exp(2)*log(5)/(x + exp((-x/3 - 1)*exp(exp(x))) + 5)

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